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We compute the phase and the modulus of an energy- and pressure-free, composite, adjoint, and
inert field φ in an SU(2) Yang-Mills theory at large temperatures. This field is physically relevant in describing part of the ground-state structure and the quasiparticle masses of excitations. The field φ possesses nontrivial S1-winding on the group manifold S3. Even at asymptotically high temperatures, where the theory reaches its Stefan-Boltzmann limit, the field φ, though strongly power suppressed, is conceptually relevant: its presence resolves the infrared problem of thermal perturbation theory.